nLab bifibration

This page is about Grothendieck fibrations that are also opfibrations. Not to be confused with two-sided fibrations nor with fibrations of 2-categories (both of which some authors also refer to as “bifibrations”).

Definition

A bifibration of categories is a functor

$$\array{ E \\ \big\downarrow \\ B }$$

that is both a Grothendieck fibration as well as an opfibration.

Therefore every morphism $f \colon b_1 \to b_2$ in a bifibration has both a push-forward $f_* : E_{b_1} \to E_{b_2}$ as well as a pullback $f^* : E_{b_2} \to E_{b_1}$.

Examples

• For $C$ any category with pullbacks, the codomain fibration $cod : [I,C] \to C$ is a bifibration.

• Dually, for $C$ any category with pushouts, the domain opfibration $dom : [I,C] \to C$ is a bifibration.

• The canonical functor Mod $\to$ CRing is a bifibration.

• The forgetful functor Top $\to$ Set is a bifibration. See also topological concrete category.

• The forgetful functor Grpd $\to$ Set is a bifibration.

• The forgetful functor Cat $\to$ Set is a bifibration.

Properties

Relation to pseudofunctors in adjoints

Proposition

Under the Grothendieck construction, the Grothendieck fibrations which arise from pseudofunctors $\mathcal{B} \longrightarrow Cat$ that factor through $Cat_{Adj}$ are equivalently the bifibrations.

Proof

A Grothendieck construction on a pseudofunctor yielding a bifibration is fairly immediately equivalent to all base change functors having an adjoint on the respective side (e.g. Jacobs (1998), Lem. 9.1.2). By the fact that $Cat_{adj} \to Cat$ is a locally full sub-2-category (this Prop.) this already means that the given pseudofunctor factors through $Cat_adj$, and essentially uniquely so.

See also Harpaz & Prasma (2015), Prop. 2.2.1.

Remark

Further factoring through ModCat $\longrightarrow Cat_{Ajd}$ hence yields bifibrations of model categories [Harpaz & Prasma (2015), Sec. 3; Cagne & Melliès (2020)]. See at model structures on Grothendieck constructions for more on this.

Relation to monadic descent

Ordinary Grothendieck fibrations correspond to pseudofunctors to Cat, by the Grothendieck construction and hence to prestacks. For these one may consider descent.

If the fibration is a bifibration, there is a particularly elegant algebraic way to encode its descent properties; this is monadic descent. The Benabou–Roubaud theorem characterizes descent properties for bifibrations.

Bifibration of bicategories

The basic theory of 2-fibrations, fibrations of bicategories, is developed in (Buckley). One (unpublished) proposal by Buckley and Shulman, for a “2-bifibration”, a bifibration of bicategories, is a functor of bicategories that is both a 2-fibration and a 2-opfibration. This should correspond to a pseudofunctor into $2 Cat_{adj}$, just as for 1-categories. There is a difference from this case, however, in that although a 2-fibration (corresponding to a “doubly contravariant” pseudofunctor $B^{coop} \to 2Cat$) has both cartesian lifts of 2-cells and 1-cells, a 2-opfibration (corresponding to a totally covariant pseudofunctor $B\to 2Cat$) has opcartesian lifts of 1-cells but still cartesian lifts of 2-cells.

Related concepts

• Grothendieck fibration

• Beck-Chevalley condition

• hyperdoctrine

• A bifibration $F:E\to B$ such that the transition functors also have right adjoints is sometimes called a trifibration (cf. Pavlović 1990, p.315) or $\ast$-bifibration.

References

Original notion and terminology of “bifibration”:

• Alexander Grothendieck, Catégories co-fibrées, catégories bi-fibrées., Section 10 in exposé VI of: Revêtements Etales et Groupe Fondamental - Séminaire de Géometrie Algébrique du Bois Marie 1960/61 (SGA 1), LNM 224 Springer (1971) [updated version with comments by M. Raynaud: arxiv.0206203]

Further early discussion (not using the terminology “bifibration”, though):

• John W. Gray, pp. 59 of: Fibred and Cofibred Categories, in: S. Eilenberg, D. K. Harrison, S. MacLane, H. Röhrl (eds.): Proceedings of the Conference on Categorical Algebra - La Jolla 1965, Springer (1966) 21-83 [doi:10.1007/978-3-642-99902-4]

Discussion in the context of the Beck-Chevalley condition:

• Duško Pavlović, Categorical Interpolation: Descent and the Beck-Chevalley Condition without Direct Images in:_Category theory Como 1990_, LNM 1488 Springer (1991) 306-325 [doi:10.1007/BFb0084229, pdf]

In the context of categorical semantics for dependent types:

• Bart Jacobs, pp. 511 of: Categorical Logic and Type Theory, Studies in Logic and the Foundations of Mathematics 141, Elsevier (1998) [ISBN:978-0-444-50170-7, pdf]

• Paul-André Melliès, Noam Zeilberger, Type refinement and monoidal closed bifibrations, (arXiv.1310.0263).

Relation to pseudofunctors with values in $Cat_{Adj}$ and $ModCat$ (cf. model structures on Grothendieck constructions):

• Alexandru E. Stanculescu, Bifibrations and weak factorisation systems, Applied Categorical Structures 20 1 (2012) 19–30 [doi:10.1007/s10485-009-9214-3]

• Yonatan Harpaz, Matan Prasma, Section 2.2. of: The Grothendieck construction for model categories, Advances in Mathematics 281 (2015) 1306-1363 [arXiv:1404.1852, 10.1016/j.aim.2015.03.031]

• Pierre Cagne, Paul-André Melliès, On bifibrations of model categories, Advances in Mathematics 370 (2020) 107205 [arXiv:1709.10484, doi:10.1016/j.aim.2020.107205]